Nuprl Lemma : rv-perp-same-norm
∀rv:InnerProductSpace. ∀x:Point. (x # 0 ⇒ (∃y:Point. ((||y|| = ||x||) ∧ (x ⋅ y = r0))))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
req: x = y,
int-to-real: r(n),
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uiff: uiff(P;Q),
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
rv-perp-1,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rv-0_wf,
ss-point_wf,
rv-norm-eq-iff,
int-to-real_wf,
rleq-int,
false_wf,
rv-ip_wf,
rnexp_wf,
le_wf,
req_functionality,
req_weakening,
rnexp-one,
rv-mul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rmul_wf,
rabs_wf,
rv-norm-nonneg,
rmul-one,
req_transitivity,
rv-norm-mul,
rmul_functionality,
rabs-of-nonneg,
rmul-zero,
rv-ip-mul2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
natural_numberEquality,
independent_pairFormation,
dependent_set_memberEquality,
because_Cache,
dependent_pairFormation,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x:Point. (x \# 0 {}\mRightarrow{} (\mexists{}y:Point. ((||y|| = ||x||) \mwedge{} (x \mcdot{} y = r0))))
Date html generated:
2017_10_04-PM-11_52_08
Last ObjectModification:
2017_03_14-PM-02_22_13
Theory : inner!product!spaces
Home
Index